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Tchebychev polynomial of second kind: tchebyshev2

tchebyshev2 takes as argument an integer n and optionnally a variable name (by default x).
tchebyshev2 returns the Tchebychev polynomial of second kind of degree n.
The Tchebychev polynomial of second kind U(n, x) is defined by:

U(n, x) = $\displaystyle {\frac{{\sin((n+1).\arccos(x))}}{{\sin(\arccos(x))}}}$

or equivalently:

sin((n + 1)x) = sin(x)*U(n, cos(x))

The U(n, x) verifies the recurrence relation:

U(0, x) = 1,    U(1, x) = 2x,    U(n, x) = 2xU(n - 1, x) - U(n - 2, x)

The polynomials U(n, x) are orthogonal for the scalar product

< f, g > = $\displaystyle \int_{{-1}}^{{+1}}$f (x)g(x)$\displaystyle \sqrt{{1-x^2}}$dx

Input :
Output :
Input :
Output :

sin(4.x) = sin(x)*(8*cos(x)3 -4.cos(x)) = sin(x)*U(3, cos(x))

giac documentation written by Renée De Graeve and Bernard Parisse